make an expensive International call
Absolutely essential to have a way to do this without worrying about cost. Most people use Skype; there are a wealth of other VOIP apps, but this is really something to get and have sorted as an "in case of emergency" backup. It is infuriating to have unlimited high-speed internet on one's phone, but be faced with paying £1 a minute to make an old-fashioned voice call...but it is what it is.
Is there some EC261/2004 compensation to be had here
It sounds like it, if BA didn't have their people ready, but how readily they will pay up depends on how the delay got coded/attributed. If, for example, the aircraft was late arriving due to inclement weather elsewhere, they may focus on this rather than the fact it was then made much later by their operational failures. Claim!